Why is it interesting?

A topos means "location," or "place"–the plural, since this is a Greek word, is topoi; but in English we often also call them "toposes"–and it is the ultimate generalization of the concept of space. […]

So you'd think that topos theory should be pretty promising as a new foundation for quantum mechanics. Indeed, it is! And the great value that topoi add here is that they allow us to get away from the–artificial, in my view–reliance on probabilities (the Copenhagen interpretation), or even more outlandish interpretations of quantum weirdness: Many Worlds, or the de Broglie-Bohm approach, or anything else. We may find a topos that describes quantum logic directly from the mathematics (where the new topos–and Linton's is a good candidate–replaces the usual mathematical universe). We now have no need to resort to "collapsing the wave function" and then relying on the mysteriously appearing "probabilities," or on any of the other devices theoretical physicists use to describe quantum behavior.

It turns out that in 1997, the idea was discovered by Chris Isham of Imperial College, London, who wrote a key article about using topos theory in quantum mechanics (you can find it on arXiv). By 2011, several articles providing more detail about how topos theory could be used to explain quantum mechanics in a natural, mathematically-powerful way were published by Isham and his colleague Andreas Döring at Imperial College (all are on arXiv), so that a virtual revolution is now taking place in the foundations of physics. http://www.science20.com/greatest_science_mysteries/buddha_topoi_and_quantum_gravity-93336

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